By Bertrand Eynard
The challenge of enumerating maps (a map is a collection of polygonal "countries" on a global of a undeniable topology, no longer unavoidably the airplane or the field) is a vital challenge in arithmetic and physics, and it has many purposes starting from statistical physics, geometry, particle physics, telecommunications, biology, ... and so on. This challenge has been studied via many groups of researchers, quite often combinatorists, probabilists, and physicists. on account that 1978, physicists have invented a style referred to as "matrix versions" to deal with that challenge, and plenty of effects were obtained.
Besides, one other very important challenge in arithmetic and physics (in specific string theory), is to count number Riemann surfaces. Riemann surfaces of a given topology are parametrized through a finite variety of actual parameters (called moduli), and the moduli area is a finite dimensional compact manifold or orbifold of complex topology. The variety of Riemann surfaces is the amount of that moduli house. extra often, a huge challenge in algebraic geometry is to signify the moduli areas, by way of computing not just their volumes, but additionally different attribute numbers known as intersection numbers.
Witten's conjecture (which used to be first proved via Kontsevich), used to be the statement that Riemann surfaces may be acquired as limits of polygonal surfaces (maps), made from a really huge variety of very small polygons. In different phrases, the variety of maps in a definite restrict, may still supply the intersection numbers of moduli spaces.
In this publication, we exhibit how that restrict occurs. The aim of this e-book is to give an explanation for the "matrix version" strategy, to teach the most effects bought with it, and to match it with equipment utilized in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions).
The publication intends to be self-contained and obtainable to graduate scholars, and offers complete proofs, numerous examples, and offers the final formulation for the enumeration of maps on surfaces of any topology. after all, the hyperlink with extra basic subject matters reminiscent of algebraic geometry, string concept, is mentioned, and particularly an explanation of the Witten-Kontsevich conjecture is provided.